arignment following Denive the for 1) \( \Delta x(10) \) 2) \( \Delta_{x}(5 x) \) 3) \( \Delta_{x}\left(10 x^{4}\right) \) 4) \( \Delta x\left(4 x^{3}+2 x^{2}+3 x+5\right) \) 5) \( D_{x}\left(10 x^{5}+4 x^{4}+2 x^{3}+4 x^{2}+10 x\right. \)

To compute the differences and derivatives, let’s break them down step by step while having some fun! 1) For \( \Delta x(10) \), since 10 is a constant, it doesn't change with respect to \( x \). Therefore, \( \Delta x(10) = 0 \) because the difference in a constant value is zero. That's a straightforward no-selector! 2) Next, \( \Delta_{x}(5 x) \) represents the change in the function \( 5x \). The change across a differential \( \Delta x \) gives \( 5\Delta x \). So, if you put it plainly: more x, more fun! 3) Moving to \( \Delta_{x}\left(10 x^{4}\right) \): this follows the same logic! It becomes \( 10\Delta x(4 x^{3}) \), resulting in \( 40x^{3} \Delta x \). So, multitudes of polynomial degrees just keep collecting changes! 4) For \( \Delta x\left(4 x^{3}+2 x^{2}+3 x+5\right) \), apply our change: this would be \( (12 x^{2} + 4 x + 3) \Delta x \). The derivatives wrap you in a puzzle, giving you math nostalgia. 5) Finally, \( D_{x}\left(10 x^{5}+4 x^{4}+2 x^{3}+4 x^{2}+10 x\right) \) gives \( 50 x^{4} + 16 x^{3} + 6 x^{2} + 8 x + 10 \). The derivative joyride takes you through the thrill of exponential growth! Math is not just numbers; it’s the exhilarating ride of finding changes and learning about the motion of curves!